Balanced clique subdivisions and cycles lengths in $K_{s, t}$-free graphs

TL;DR

This paper improves bounds on clique subdivisions and cycle lengths in $K_{s,t}$-free graphs, using sublinear expansion and structural techniques.

Contribution

It provides a stronger result on balanced clique subdivisions and cycle length reciprocals in $K_{s,t}$-free graphs, confirming conjectures and improving previous bounds.

Findings

Balanced clique subdivisions of order $igOmega(d^{s/(2(s-1))})$ exist in $K_{s,t}$-free graphs.

Lower bounds on the sum of reciprocals of cycle lengths are improved to $(s/(2(s-1)) - o_d(1)) imes ext{log } d$.

Both results use graph sublinear expansion and novel structural methods.

Abstract

Let $ t\ge s\ge2$ be integers. Confirming a conjecture of Mader, Liu and Montgomery [J. Lond. Math. Soc., 2017] showed that every $K_{s, t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $\Omega(d^{\frac{s}{2(s-1)}})$ vertices. We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times. In 1975, Erd\H{o}s asked whether the sum of the reciprocals of the cycle lengths in a graph with infinite average degree $d$ is necessarily infinite. Recently, Liu and Montgomery [J. Amer. Math. Soc., 2023] confirmed the asymptotically correct lower bound on the reciprocals of the cycle lengths, and provided a lower bound of at least $(\frac{1}{2} -o_d(1)) \log d$. In this paper, we improve this low bound to $\left(\frac{s}{2(s-1)} -o_d(1)\right) \log d$ for $K_{s, t}$-free graphs. Both proofs of our results use the graph sublinear expansion property as well as some novel structural techniques.